1. Technical Field of the Invention
The present invention relates to an implicit function rendering method of a nonmanifold, a direct drawing method of an implicit function curved surface and programs thereof.
2. Description of the Related Art
When a surface shape of a three-dimensional object is rendered by a computer, parametric rendering is widely used, which defines the shape by a coordinate value of the surface of the object. As examples, free curved surface rendering used by a computer-aided design (CAD), polygon rendering used by a virtual reality model and in an entertainment field, etc. can be cited. Parametric rendering has a feature that a data structure is intuitive and easy to understand, a shape can be rendered by a relatively small amount of data, and use of high-speed drawing hardware is easy.
On the other hand, when a process such as deformation of a surface shape or a set operation is carried out, it is often convenient to use implicit function rendering, which uses a function of an implicit form to indirectly render a curved surface. This rendering is used in fields of modeling, shape processing, physical simulation, etc. This rendering form has a feature that it facilitates a complex process, because of its simple data structure and its suitability for hierarchization or parallelization of the process. Parametric rendering and implicit rendering of a curved surface have advantages and disadvantages. Capability of selecting the form in accordance with a process is preferably provided, and, in such a case, a need arises to convert both forms.
As prior art documents concerning the present invention, the following can be cited.
[Non-patent Document 1]
    J. Bloomenthal, Polygonization of implicit surfaces, Computer Aided Geometric Design, 5:341–355, 1988.[Non-patent Document 2]    J. Bloomenthal, Introduction to Implicit Surface, Morgan Kaufmann Publishers, Inc., 1997.[Non-patent Document 3]    J. Bloomenthal and K. Ferguson, Polygonization of non-manifold implicit surfaces, In SIGGRAPH '95, pages 309–316, 1995.[Non-patent Document 4]    M. Brady, K. Jung, H. T. Nguyen and T Nguyen, Two-phase perspective ray casting for interactive volume navigation, In Visualization 97, pages 243–56, 1997.[Non-patent Document 5]    K. Engel, M. Kraus and T. Ertl, High-quality pre-integrated volume rendering using hardware-accelerated pixel shading, In Eurographics/SIGGRAPH Workshop on Graphics Hardware '01, pages 9–16, 2001.[Non-patent Document 6]    H. C. Hege, M. Seebas, D. Stalling and M. Zockler, A generalized marching cubes algorithm based on non-binary classifications, Technical report, Konrad-Zuse-Zentrum fur Information stechnik (ZIB), 1997.[Non-patent Document 7]    Philippe Lacroute and Marc Levoy, Fast volume rendering using a shear-warp factorization of the viewing transformation, In SIGGRAPH '94, pages 451–458, 1994.[Non-patent Document 8]    W. E. Lorensen and H. E. Cline, Marching cubes: a high resolution 3d surface reconstruction algorithm, In SIGGRAPH '87, pages 163–169, 1987.[Non-patent Document 9]    J. Rossignac and M. O'Connor, SGC:A dimension independent model for pointsets with internal structures and incomplete boundaries, In Geometric Modeling for Product Engineering, pages 145–180, 1990.[Non-patent Document 10]    H. Tuy and L. Tuy, Direct 2d display of 3d objects, IEEE Mag, Computer Graphics and Applications, 4(10):29–33, 1984.[Non-patent Document 11]    A. P. Witkin and P. S. Heckbert, Using particles to sample and control implicit surfaces, In SIGGRAPH '94, pages 269–278, 1994.[Non-patent Document 12]    Shuntaro Yamazaki, Kiwamu Kase and Katsushi Ikeuchi, Nonmanifold implicit surfaces based on discontinuous implicitization and polygonization, In Geometric Modeling and Processing, pages 138–146. IEEE, July 2002.
Many methods have been presented concerning conversion of implicit function rendering into parametric rendering. Especially, conversion into a triangle mesh can be carried out stably and at a high speed. However, methods for converting parametric rendering into implicit function rendering do not always provide good results. In the conventional implicit function rendering, a surface to be handled is limited to two kinds of manifolds because of use of a continuous real valued function for defining a function field. Thus, if there are nonmanifold characteristics in an entry, for example, if there is a boundary on a curved surface and a branch on a surface, there is a problem that these portions are lost to greatly change shapes.
On the other hand, implicit function rendering itself has sufficient information to represent a curved surface. However, since a value to be held as data is not directly connected to a geometric shape of the curved surface, the surface must be made visible in order to check a real shape. When shape designing is carried out by using the implicit function curved surface, however, remeshing must be carried out because of a change in a function field made by a process such as deformation sequentially updated during the process of surface mixing or deformation, and, consequently, the process takes time. Additionally, as a surface shape becomes more complex, the number of formed polygons is increased, creating a problem of a slow drawing speed.